keyholes and mimo channel modelling

KEYHOLES AND MIMO CHANNEL MODELLING

Alain SIBILLE [email protected]

ENSTA32 Bd VICTOR, 75739 PARIS cedex 15, FRANCE

COST 273, Bologna meetingCOST 273, Bologna meeting

Outline

Keyholes in MIMO channels viewed as the result of diffraction

Outline


Channel modelling with multipath junctions in a smallSize, uncoupled sensors antenna approximation

Outline



How to include inter-sensors coupling

Outline




towards a stochastic MIMO channel model

Outline




towards a stochastic MIMO channel model

Conclusion

333131

232221

131211

321

3

2

1

BABABA

BABABA

BABABA

KAAAK

B

B

B

H

0)()()( 23112311 BAEBAEBABAE : uncorrelated (complex) entries

K

A1A2

A3

B1B2

B3

Rank(H)=1 (two null coefficients of characteristic polynomial)

Slit transmittance

Keyholes : The concept of « Keyholes » has been suggested by Chizhik in order to hightlight the imperfect correspondence between rank and correlation. In a keyhole, the channel matrix has uncorrelated entries, but its rank is one. Such keyholes have therefore intrinsically a small capacity, even in a rich scattering environment.

1D channel

A simple numerical example of keyhole using Kirchhoff diffraction:

Large slit: no diffraction

Rx Tx

Keyholes in MIMO channels

A simple numerical example of keyhole using Kirchhoff diffraction:

Large slit: no diffraction Narrow slit: diffraction and multipath junction 1 3

Rx Tx

Rx Tx

)exp()exp()exp()exp(..),,(3

1,1tjrij

jiiijjitr rkjrkjljkjklKTRrrH

junction


Kij computed by Kirchhoff diffraction

2.5 3 3.5 40

0.2

0.4

0.6

0.8

cum

ulat

ed p

roba

bilit

y

case A

wide slitlittle correlation : 3 DFSV: -7, +4.8, +7.6 dB

HH

n

SNRIC

tnr

.detlog2


capacity (b/s/Hz)

H: (normalized) channel transmission matrix

nt=3: number of Tx, Rx radiators

SNR = 3 dB

Space-variant stochastic ensemble

2.5 3 3.5 40

0.2

0.4

0.6

0.8

cum

ulat

ed p

roba

bilit

y

case A

case B


B: narrow slit, little correlation : 1 DFSV: -47, -28, +9.5 dB


capacity (b/s/Hz)

2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

cum

ulat

ed p

roba

bilit

y

case A

case C case B


C: narrow slit, strong correlation : 1 DFSV: -111, -41, +9.5 dB



capacity (b/s/Hz)

2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

capacity (b/s/Hz)

cum

ula

ted

pro

babi

lity

5 2

Slit width in units of

Keyholes in MIMO channels: capacity vs. Slit width

2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

capacity (b/s/Hz)

cum

ula

ted

pro

babi

lity

5 2 0.25 0.5



2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

capacity (b/s/Hz)

cum

ula

ted

pro

ba

bili

ty

5 2 1 0.25 0.5



2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

capacity (b/s/Hz)

cum

ula

ted

pro

ba

bili

ty

5 2 1 0.25 0.5


When d< ~ /2 all incoming waves are diffracted into all exiting waves through a 1-dimensional channel


2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

capacity (b/s/Hz)

cum

ula

ted

pro

ba

bili

ty

5 2 1 0.25 0.5


When d< ~ /2 all incoming waves are diffracted into all exiting waves through a 1-dimensional channel

When d>~2 transmission through the slit occurs through multiple modes and evanescent states and results in greater 3 dimensional effective channel


333131

232221

131211

321

3

2

1

BABABA

BABABA

BABABA

KAAAK

B

B

B

H

0)()()( 23112311 BAEBAEBABAE : uncorrelated (complex) entries

K

A1A2

A3

B1B2

B3

Rank(H)=1 (two null coefficients of characteristic polynomial)junction

Keyholes : correlations or no correlations ?

333131

232221

131211

321

3

2

1

BABABA

BABABA

BABABA

KAAAK

B

B

B

H

0)()()( 23112311 BAEBAEBABAE

0

.2

1

2

121

21112111

AEAEBEBE

BAEBAEBABAE

: uncorrelated (complex) entries

: correlated amplitudes

K

A1A2

A3

B1B2

B3

Rank(H)=1 (two null coefficients of characteristic polynomial)junction

Keyholes : correlations or no correlations ?

2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

capacity (b/s/Hz)

cum

ulat

ed p

roba

bilit

y

case A

case D

case C case B


C: narrow slit, strong correlation : 1 DFSV: -111, -41, +9.5 dB


narrow slit, strong correlation, one random phase: 2 DFSV: -40, -3.9, +9.4 dB

fading


)()()()( Ttr aWaH

Small antenna, uncoupled sensors approximation: {ar , at : steering matrices for N DOAs and M DODs (nrXN , MXnt )

W : wave connecting matrix (NXM) : complex attenuations from all DODs to all DOAs

W is in general rectangular in the presence of path junctions (diffraction, refraction …)

Rank(H) Min(nr , nt , N , M)

All MIMO properties determined

by the geometry of sensors and by W (DOD, DOA, complex amplitudes)

)()()()( Ttr aWaH

MIMO channel modelling

DOD

00000

00000

0000

0000

00000

X

X

XX

XX

X

DOA

Example: channel correlation matrix(US approximation, spatial averaging)

ii

iiii

iinmmn WW,

2

,,

2

, /expR tnntitrmmrri rrkjrrkj

)()()()()()( ** HE trTtr aWaaWaR

n’

nm’

m

DOA DOD

MIMO channel modelling

Receiver sensors positions Transmitter radiators positions

Rx Tx

ik

Steering matrix ar (nrXN)

)exp(,,ra rij rkjji

Uncoupled sensors

MIMO channel modelling : case of coupled sensors

)()()()( Ttr aWaH

ik

ik

Steering matrix ar (nrXN) Complex gain matrix Gr (nrXN)

)exp(,,ra rij rkjji

)(,rG,,rG jkiji

Uncoupled sensors Coupled sensors

MIMO channel modelling : case of coupled sensors

)()()()( Ttr aWaH )()(W)()(H T

tr GG

specification of Tx and Rx antennas, either through steering matrices (uncoupled sensors) or through complex gain matrices

Towards a stochastic MIMO channel model


double directional model of emitted and received waves, specifying the statistical laws of angular distributions on both sides of the radio link




statistical model for the wave connecting matrix , specifying the distribution of complex entries of the matrix, especially the number of non zero entries for the various columns or lines and their relative amplitudes.




statistical model for the wave connecting matrix , specifying the distribution of complex entries of the matrix, especially the number of non zero entries for the various columns or lines and their relative amplitudes.

statistical model for the distribution of delays involved in the non zero entries of )(W


Introduction of artificial junctions to reduce DOA/DOD number: depend on maximum antenna size/angular resolution

Tx Rx

MIMO channel model simplification


Tx Rx

Tx Rx



Limitation on the dynamic range of wave amplitudes: substitution of numerous small amplitude waves by one or a few Rayleigh distributed waves of random DOA/DOD.


Tx Rx

Tx Rx

X

Y

Y

X

.

.

.

• look for a differing number of DOAs and DODs

• look for several path delays for the same DOA (or DOD)

Double directional channel measurements and junctions ?

Analysis of keyholes through Kirchhoff diffraction: continuous variation of channel matrix effective rank with slit width

Small antenna approximation yields a MIMO channel description entirely based on DOA, DOD and antennas geometry

Junctions in multipath structure is responsible for the rectangular or non diagonal character of the « wave connecting matrix »

Coupling between sensors readily incorporated

Stochastic channel model. Simplifications as a function of precision requirements

May feed simpler MIMO channel models with environment dependent channel correlation matrices

Conclusion

keyholes and mimo channel modelling

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